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Use the method of reduction of order to find a second linearly independent solution 32 (a) (b) x²y" + xy - 9y=0; x > 0, y₁(x) = x³. Ay" - 4y + y = 0; 3₁(x)=e¹/2.

Use the method of reduction of order to find a second linearly independent solution 32 (a) (b) x²y" + xy - 9y=0; x > 0, y₁(x) = x³. Ay" - 4y + y = 0; 3₁(x)=e¹/2.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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HW7P6

6. Use the method of reduction of order to find a second linearly independent solution \( y_2 \): (a) \[ x^2 y'' + xy' - 9y = 0; \, x > 0, \, y_1(x) = x^3. \] (b) \[ 4y'' - 4y' + y = 0; \, y_1(x) = e^{x/2}. \]
Transcribed Image Text:6. Use the method of reduction of order to find a second linearly independent solution \( y_2 \): (a) \[ x^2 y'' + xy' - 9y = 0; \, x > 0, \, y_1(x) = x^3. \] (b) \[ 4y'' - 4y' + y = 0; \, y_1(x) = e^{x/2}. \]
Expert Solution
Part(a) Step 1

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.

x^2y''+xy'-9y=0

divide both sides by x2

y''+\frac{1}{x}y'-\frac{9}{x^2}y=0

compare this DE with y''+ p(x)y'+q(x)y=0

so here we have

p(x)=\frac{1}{x}

and the given solution is 

y_1=x^3

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